If $\int e^x\ dx = e^x + C$ What does the $\int e^{2x}=$ If $\int e^x\ dx = e^x + C$ What does the $\int e^{2x}=$ Please accompany answer with rule.

Substituting $2x$ with $y$ : $$\int e^{2x}dx = \int e^ydx$$ $$y = 2x \iff dy = 2dx \iff dx= \frac{1}{2}dy$$

Hence, $$\int e^{2x}dx = \int e^y \frac{1}{2}dy = \frac{1}{2} (e^y + C) = \frac{1}{2} (e^{2x} + C) $$

The rules used are integration by substitution and linearity of integration.